so the time resolution becomes independent of the threshold. This can be understood when looking at Fig. REF a and it is a well-established fact for detectors like Resistive Plate Chambers [1]}, where avalanche fluctuations dominate the signal characteristics [2]}, [3]}: scaling the threshold by a constant \(c_1\)  will just shift the time response function by \(\Delta t =( \ln c_1)/\lambda _t\)  without altering its shape. If we are not interested in the absolute time of the threshold crossing but just the time variations we can arbitrarily set \(n=1\)  and have the time response function
\(\rho (t) = \frac{\lambda _t}{\Gamma (A)} \exp \left[ -A \lambda _t t - e^{-\lambda _t t} \right] \qquad \int _{-\infty }^\infty \rho (t) dt = 1\) 
